KC Sinha Mathematics Solution Class 12 Chapter 12 द्वितीय कोटि का अवकलज (Second Order Derivative) Exercise 12.1 (Q4-Q6)

Exercise 12.1

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Question 4

यदि(If) y=sinx(sinx) , दिखाएँ कि(show that) $\frac{d^{2} y}{d x^{2}}+\tan x \frac{d y}{d x}+y \cos ^{2} x=0$
Sol :
y=sin(sinx)

Differentiating w.r.t x

$\frac{d y}{d x}=\cos (\sin x)\times \cos x$

$\cos(\sin x)=\frac{1}{\cos x} \cdot \frac{dy}{d x}$

Again ,Differentiating w.r.t x

$\frac{d^{2} y}{d x^{2}}=-\sin (\sin 3 x) \cdot \cos x \cdot \cos x+\cos (\sin x) \times(-\sin x)$

$\frac{d^{2} y}{d x^{2}}=-y \cos ^{2} x \cdot+\frac{1}{\cos x} \cdot \frac{d y}{d x}(-\sin x)$

$\frac{d^{2} y}{d x^{2}}=-y \cos ^{2} x-\tan x \frac{d y}{dx}$

$\frac{d^{2} y}{d x^{2}}+\tan x \frac{d y}{d x}+y\cos^{2} x=0$

Question 5

यदि(If) $y=[\log (x+\sqrt{x^{2}+a^{2}})]^{2}$ , सिद्द करे कि(prove that)  $\left(x^{2}+a^{2}\right) y_{2}+x y_{1}=2$
Sol :
$y=\left[\log(x+\sqrt{x^{2}+a^{2}})\right]^{2}$

Differentiating w.r.t x

$y_{1}=2\left[\log (x+\sqrt{x^{2}+a^{2}}) \cdot \frac{1}{x+\sqrt{x^{2}+a^{2}}} \times\left(1+\frac{1}{2 \sqrt{x^{2}+a^{2}}} \times 2 x\right.\right)$

$y_{1}=2 \frac{\left[\log(x+\sqrt{x^{2}+a^{2}}\right]}{x+\sqrt{x^{2}+a^{2}}}\left[\frac{\sqrt{x^{2} + a^{2}}+x}{\sqrt{x^{2}+a^{2}}}\right]$

$y_{1}=\frac{2[\log (x+\sqrt{x^{2}+a^{2}})]}{\sqrt{x^{2}+a^{2}}}$

Again ,Differentiating w.r.t x

$y_{2}=\frac{2 \cdot \frac{1}{x+\sqrt{x^{2}+a^{2}}}\left[1+\frac{1}{2 \sqrt{x^{2}+a^{2}}} \times 2 x\right] \cdot \sqrt{x^{2}+a^{2}}-2[\log(x+\sqrt{x^2+a^2})]\times \frac{1}{2\sqrt{x^2+a^2}}\times 2x }{(\sqrt{x^{2}+a^{2}}]^{2}}$

$y_{2}=\frac{\frac{2}{x+\sqrt{x^{2}+a^{2}}}\left[\frac{\sqrt{x^{2}+a^{2}}+x}{\sqrt{x^{2}+a^{2}}}\right] \cdot \sqrt{x^{2}+a^{2}}-\frac{2 \times \log (x+\sqrt{x^{2}+a^{2}}}{\sqrt{x^{2}+a^{2}}}}{x^{2}+a^{2}}$

$\left(x^{2}+a^{2}\right) y_{2}=2-xy_{1}$

$\left(x^{2}+a^{2}\right) y_{2}+x y_{1}=2$

Question 6

यदि(If) $y=\left(1-x^{2}\right)^{3 / 2}$ ,  दिखाएँ कि (show that) $\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}+3 y=0$
Sol :
$y=\left(1-x^{2}\right)^{\frac{3}{2}}$

Differentiating w.r.t x

$\frac{d y}{d x}=\frac{3}{2}\left(1-x^{2}\right)^{\frac{1}{2}} \cdot[0-2 x]$

$\frac{d y}{dx}=\frac{3}{2}\left(1-x^{2}\right)^{\frac{1}{2}} \cdot(-2 x)$

$\frac{dy}{d x}=-3 x\left(1-x^{2}\right)^{\frac{1}{2}}$

Again ,Differentiating w.r.t x

$\frac{d^{2} y}{d x^{2}}=-3 \cdot\left(1-x^{2}\right)^{\frac{1}{2}}+(-3 x) \frac{1}{2}\left(1-x^{2}\right)^{-\frac{1}{2}} \times(-2 x)$

$\frac{d^{2} y}{d x^{2}}=-3\left(1-x^{2}\right)^{\frac{1}{2}}+\frac{3 x^{2}}{\left(1-x^{2}\right)^{\frac{1}{2}}}$

Multiplying by $\left(1-x^{2}\right)$ in both sides

$\left(1-x^{2}\right) \cdot \frac{d^{2} y}{d t^{2}}=-3\left(1-x^{2}\right)^{\frac{3}{2}}+3 x^{2} \cdot\left(1-2^{2}\right)^{\frac{1}{2}}$

$\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}=-3 y-x\left[-3 x\left(1-x^{2}\right)^{\frac{1}{2}}\right]$

$\left(1-x^{2}\right) \cdot \frac{d^{2} y}{d x^{2}}=-3 y-x \frac{d y}{d x}$

$\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}+3 y=0$